Questions tagged [plane-geometry]

Plane Geometry is about flat shapes like lines, circles and triangles , shapes that can be drawn on a piece of paper

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4
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0answers
64 views

Zero-dimensional functions in the plane

Is the following true? Conjecture. Let $\varphi:C\to [0,\infty)$ be an upper semi-continuous function, where $C\subseteq \mathbb R$ is a Cantor set. Let $X$ be a zero-dimensional subset of the graph ...
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Generalized figures of constant width

Is it known which plane figures $Q$ can rotate touching three given circles $A$, $B$, and $C$? This question was asked by Lazar Lyusternik in 1946, there is only one reference to this paper that ...
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154 views

Hypothesis: An injection from polygons into $SO(2) \times S_n$

I have stumbled upon a possible representation of polygons by a concise description of their behaviour under rotation. I would like to know more about it and, obviously, if it is actually a bijection. ...
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1answer
122 views

Orientations of triples of points in the plane

Given a finite indexing-set $I$ and a collection $P = \{P_i: \ i \in I\}$ of points in the plane no three of which are collinear, let $I_{(3)}$ denote the set of ordered triples of distinct elements ...
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Continuous map from $\mathbb C^4$ to $\mathbb R$ that changes sign under circular permutation of coordinates and that is $0$ only for squares

Does there exist a continuous map $f$ from $\mathbb C^4$ to $\mathbb R$ such that: i) there exists four distinct complex numbers $a$, $b$, $c$, $d$, s.t. $f(a,b,c,d)f(b,c,d,a)<0$ ii) for every $(...
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1answer
29 views

On comparing planar convex regions of equal perimeter and area

Definitions: The Hausdorff distance between two point sets is the greatest of all the distances from a point in one set to the closest point in the other set. Given two planar convex regions $C_1$ ...
2
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1answer
76 views

Conformal isomorphism uniquely determined by boundary identification?

Let $\Gamma$ be a smooth Jordan arc, and let $\Phi \colon \hat{\mathbb C} \backslash \overline{\mathbb D} \to \hat{\mathbb C} \backslash \Gamma$ be a conformal isomorphism that fixes the point at $\...
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1answer
201 views

Minimum area of the convex hull of the union of a parallelogram and a triangle

This question is somewhat dual to my previously stated question about Maximum area of the intersection of a parallelogram and a triangle, where the triangle and parallelogram each is assumed to be of ...
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2answers
286 views

On 'fair bisectors' of planar convex regions

Definitions (https://www.ias.ac.in/article/fulltext/pmsc/122/03/0459-0467): Given a planar convex region $C$ (could be smooth or polygonal), an area bisector of $C$ is any line that partitions $C$ ...
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1answer
201 views

Johnson-Lindenstrauss lemma preserves angles

Edit (April 1, 2020): I formalized the original question (from the reference below) wrong. Please see @AndreasBlass correction. Also, all the suggestions proposed so far doesn't seems to be related ...
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Convergence in the Caratheodory sense and Hausdorff sense

Among Jordan domains, I understand that Caratheodory convergence is weaker than Hausdorff convergence. But if a sequence of Jordan domains all have rectifiable boundary whose arc length are all $L$, ...
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1answer
853 views

How can we find n points on a plane so that as many pairs of points as possible have the same distance?

There are $n$ points on the plane, and we need to maximize the number of pairs of points which have the same Euclidean distance.
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2answers
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On conics curves and increasing unions of ellipses

It is easy to see that the epigraph of a parabola, i.e. the set $ \\{(x,y)\in \mathbb R^2, y> x^2\\} $ is a countable increasing union of ellipses in the sense that $$ \\{(x,y)\in \mathbb R^2, y&...
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The status of the journal “Forum Geometricorum”

The online journal Forum Geometricorum is a sort of central organ of elementary geometry (mainly triangle geometry and related topics). It has been published regularly since 2000 but seems to have ...
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463 views

Geometry of Level sets of elliptic polynomials in two real variables

Updated: A polynomial $P(x,y)\in \mathbb{R}[x,y]$ is called an elliptic polynomial if its last homogeneous part does not vanish on $\mathbb{R}^2\setminus\{0\}$.The two answers to this post provide ...
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2answers
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Finding a not too slim triangulation with prescribed vertices on $\mathbb R^2$

Let us fix a constant $r>1$. Let $d(x,y)$ denote the distance between points $x,y\in \mathbb R^2$. Suppose we have a discreet subset $X\subset \mathbb R^2$ such that 1) For any two points $x,x'\...
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Complexity of tour-expansion heuristic for the planar Euclidean TSP

This is a followup question to this one: Computational Geometric Aspects of Greedy Tour Expansion. Assume that the candidate point, whose insertion into current incurs the least tour-length increase, ...
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1answer
142 views

Does any real projective plane incidence theorem follow from axioms?

Is it known whether any projective geometry statement that holds true in the real projective plane (equivalently, can be deduced from Hilbert axioms) follows from the standard projective axiomatics? ...
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2answers
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Algorithm for reporting all triangles with unique interior point

What is known about the complexity of and/or practical algorithms for reporting all triplets of points from finite set of at least four points of which no three are collinear in the Euclidean plane, ...
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3answers
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Calculating radii allowing for circular placement of polygonal linkage's joints

Given a planar polygonal linkage defined by a sequence of $n$ hinge joints $(j_0,\,\cdots,\,j_{n-1},j_n = j_0)$ with links of fixed lengths $\lbrace\|j_{k+1}-j_k\|=d_k\ |\ 0\le k\lt n\rbrace$ between ...
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1answer
287 views

Is the Wikipedia depiction of the ergosphere of a Kerr black hole a Cassini oval?

Disclaimer: this a cross post from MSE, where this question was asked on November 4th 2019 and has so far received no upvote, no comment and no answer whatsoever. Glancing at https://en.wikipedia.org/...
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Area of Disc that Intersects Another under Smooth Flow

The following question can be asked in any $\mathbb{R}^n$ for $n > 1$, but the case of interest is (thankfully) the case $n = 2$. The formulation of the problem with discs isn't actually critical ...
3
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1answer
397 views

Distance between point inside a triangle and its vertices [closed]

How to determine the distance between an arbitrary point inside a triangle and its vertices if side lengths are given. Is there any correlation between these distances or their sum and the lengths of ...
3
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1answer
92 views

Large class of curves which only intersect each other finitely many times

I am trying to find a large subset of piecewise-differentiable plane curves of finite length (subsets of $\mathbb{R}^2$) with the following property: For any pair $\gamma_1, \gamma_2$ of curves in ...
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1answer
182 views

Nowhere compact subsets of the plane

Suppose $X\subseteq \mathbb R^2$ is nowhere compact ($X$ has no compact neighborhood) and non-empty. Can $X$ be densely embedded into the plane? In other words, is there a dense set $X'\subseteq ...
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1answer
151 views

Triangle angle bisectors, trisectors, quadrisectors,

With the triangle angle bisector theorem and Morely's trisector theorem as background, are there any pretty theorems known for triangle $n$-sectors, $n > 3$? For example, angle quadrisectors? The ...
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On convex regions containing (and contained within) a given triangle

Given an arbitrary triangle T. How does one find the convex region C_M of largest area containing T such that T is also the largest area triangle that is contained within C_M? Guess: for any T, ...
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2answers
91 views

Smallest triangles that contain 2D convex regions with reflection symmetry

Given any 2D convex region $C$ with a mirror symmetry. Two pairs of questions: We need to find the smallest area (likewise, smallest perimeter) triangle that contains $C$. Is it sufficient to only ...
3
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1answer
162 views

Please identify this triangle septic

Let $ABC$ a triangle in the plane, but $D$ a point in (R3) space, such that the angles $\phi=ADB=BDC=CDA$ are equal. Let $E$ be the footpoint of $D$ in $ABC$. $E(\phi)$ describes a (irreducible) ...
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227 views

Geometric interpretation of metric [closed]

For metric in $\mathbb R^2$ (polar coordinates) Pythagorean triangle elements can be visualized in a differential form as seen at top yellow triangle. This happens for metric: $$ ds^2= dr^2 +(r d \...
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1answer
98 views

Find the smallest circumference of a figure containing n squares [closed]

So there's a figure which contains n squares of 1 x 1, and I have to find the smallest circumference possible. I don't know if there's an algorithm behind this, I've been stuck on this for two hours ...
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1answer
394 views

To minimize the Hausdorff distance between convex polygonal regions

Definition: The Hausdorff distance is the greatest of all the distances from a point in one set to the closest point in the other set. Question: Given two convex polygonal regions P1 and P2 on the ...
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1answer
113 views

Minimizing the number of segments in drawings of planar graphs

Every planar graph has at most $3n-6$ edges, where $n$ is the number of vertices. Moreover, every planar graph can be drawn with straight-line edges in the plane, without crossings. For example, for ...
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Trapping lightrays under nonstandard reflections and/or paths

Almost every version of trapping lightrays with mirrors is either resolved---usually negatively---or open: "It is unknown whether one can construct a polygonal trap for a parallel beam of light": ...
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157 views

How to find a point on a line that minimizes sum of distances from three given points?

Let there be given three points $x_1, x_2, x_3$ and a line $l$ on a plane. Does there exist an explicit method of finding such a point $p$ on $l$, that sum of distances of $p$ from $x_1, x_2, x_3$ is ...
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1answer
228 views

What is the symmetry group of this configuration?

This configuration appear as problem 3845 in Crux Mathematicorum. I see it is very beautiful. This configuration are generalization of Pascal theorem and Brianchon theorem: Consider six points $A_1$, ...
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On non-convex polygons that tile convex polygons

Polyominoes are rectilinear polygons - all angles are 90 or 270 degrees. It is well-known that among non-convex polyominoes are several whose copies can neatly tile rectangle. And for several such '...
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3answers
468 views

Two queries on triangles, the sides of which have rational lengths

Let us define a "rational triangle" as one in the Euclidean plane, with lengths of all sides rational. We are aware that a positive integer is called "congruent" only if it is the area of a right ...
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Visibility in a growing orchard

This is a variant on Polya's orchard problem.1,2 Suppose trees are planted randomly in the plane. The question is: How many trees are visible from the origin as their radii grow? More precisely, ...
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Graphs determined by monohedral, edge-to-edge tilings of the plane

Let $\cal T$ be a monohedral, edge-to-edge tiling of the plane, with prototile $T$ a simple polygon, and with one edge $e^*$ of $T$ distinguished. Associate a graph $G=G_{\cal T}$ with $\cal T$ as ...
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2answers
188 views

Geometric dissection theory

A few days ago, i realized that one way to prove the Pythagorean Theorem is to dissect the given right-angled triangle into 2 triangles similar to it, and apply well-known properties of ratios of ...
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0answers
142 views

Hyperbolic Intercept (Thales) Theorem

Is there an Intercept theorem (from Thales, but don't mistake it with the Thales theorem in a circle) in hyperbolic geometry? Euclidean Intercept Theorem: Let S,A,B,C,D be 5 points, such that SA, SC, ...
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1answer
356 views

Yau's problem: Construct a triangle given a side, an angle, and an angle bisector

In Shing-Tung Yau's autobiography The Shape of a Life, he mentions a problem that he came up with as a teenager. Suppose you know the length of one side of a triangle, one angle, and the length of ...
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2answers
173 views

A non-Borel union of unit half-open squares

On the complex plane $\mathbb C$ consider the half-open square $$\square=\{z\in\mathbb C:0\le\Re(z)<1,\;0\le\Im(z)<1\}.$$ Observe that for every $z\in \mathbb C$ and $p\in\{0,1,2,3\}$ the set $...
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2answers
1k views

Packing rectangles: Does rotation ever help?

Dominic van der Zypen posed an interesting Box stacking problem. This is a spin-off question. Let a collection of rectangles $r_1,\ldots,r_n$ be given by their side lengths in $\mathbb{R}$. Let $R$ ...
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1answer
282 views

Fractal plane continuum with order $\aleph_0$?

Continuum means compact and connected. Define the order of a point $x$ in a continuum $X$ to be the least cardinal $\alpha$ such that $X$ has a neighborhood base of open sets at $x$ with no more ...
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1answer
139 views

First-order logic of projective planes over fields [closed]

Suppose $\mathbb{P}^2(k)$ is a projective plane defined/coordinatized over a commutative field $k$. Is the first-order logic of the plane completely determined by the first-order logic of $k$ ? (In ...
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1answer
136 views

What curve of positive curvature minimizes distance from the origin, given length and total curvature?

Let $\textit{F}$ be the family of $C^1$ curves in $\mathbb{R}^2$ of fixed length $\bar{l}$ and fixed tangent's turning angle $\bar{k}$. What are the curves of positive curvature in $\textit{F}$ ...
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1answer
105 views

6-periodic billiards trajectory in acute triangle

We can construct a 3-periodic billiards trajectory in an acute triangle in a classical geometric way, say taking the altitudes. Is there a similar way to construct a 6-periodic billiards?
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What is the locus defined by those equations?

I would like to know what is the locus of $x \in \Bbb R_+^n$ ($n=2$ would already be fine) defined by $\sum a_i \cdot x_i$ s.t. $a_i+\epsilon \geq 0$, $\epsilon \in \Bbb R$. I know that if $\...

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